[PDF] Junior Section - First Round - SMO Singapore Mathematical Olympiad 2016.pdf | Plain Text

Singapore Mathematical Society Singapore Mathematical Olympiad (SMO) 2016 Junior Section (Round 1) T esday, 31 May 2016 Instructions to contestants Ansuer ALL 35 questions. Enter sau ansuers on the anslner sheet prauided.. i::,.::'":':!!!::'!:": .v"'"i:'": pntc'1 lour a*rer on the msuer sheet blt shadins the oLtDttl- anl0 n rQ lhp tptlpt lA. B C- D or C..atpspon|.na,",n, ,_.,"., ,,r,."r. !:_l!:.:t!":,:0":i*.istions, urite uour ansuer an the anslrer sheet an(t. sha.te thc. ap prapriate bubbte betow gour ansuer. No steps are neetl,etl. ta justily yaLr ansueru. Each q esti,on arries 1 marh. No calculators are altouetL. Thraughout thi.s paper, Iet lxl derrcte the greatest ezampte, 12.1): 2, ls.sl = s. integet less than or equat ta r. F.)r Throushaut this paper, Iet [A1A, ... A.) denote the area af the patsson A1A2. . . A-. Th'",ghout tt.' Doppt t.t atr-n : -.J,nat" on n-d,g,t, r,-b.t Lt,tt, tt "Jq,." n the carrespondtns positian, ,i.e. a;!a;_t . . aa : o,, ,t1.1, + o," rti:; +".".. i")in,. PLEASE DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SO 0930-1200 hrs 9. t0 Supported by Minisiry of Education Sponsored by lMicron Technology 45r7tcrost

MulLiple Choice Quesr ions 1. We know that I2:trIE and I3:YOU. If each distinct lefter represents a unique digit suc_h that lhe arithmetic holds, whai is the value of E? (A) 4 (B) 5 (c) 6 (D) B (E) e 2. Given three integer.S, we form another three integers by adding the me:ll of any two of them to the third. If the three nes' intege$ formed are 3J, 35 and 40. find the mean of .the original three integels. (A) i8 (B) 27 (C) 30 (D) 36 (E) None of the above 3. Let ABCD be a rectangle, -R be a poinr on BC such that 6t : 2EC, alld -F. be a point on ,48 such tLat ,4-f : 3,FE. If the arca ol ABCD is 1200. whai is rhe are.. of rfe quadrilateral t-4-4C? (A) 100 (B) 300 (c) 350 (D) 100 (E) None o{ the above ,1. The dia.gram sho!\'s a square,4BCD ilscribed in a semicirc]e wjth cenrre O and another square PQfiS inscribed in the eniirc circle with the same centrc. If the area of 7IBCD is 16, find the area of PQ-R,9. 5. ,43CD is a parallelogram. .q is a point on the sidc r4R such tha,t the ratio of rhe arca of the quaddlat€ral , EC.lf to the axea of the triangle ,4EC is 7 : 5. The ratio of,4, : tB js (A) 20 (B) 24 (c) 32 (D) 40 (E) 48 (D) 3i5 2 (A) 2:3 (B) 2:7 (c) 3:4 (o) 5:z

6. In the diagram , is the midpoirt of AB, EF :2CF arLd AF is parallel to D_8. Given the area of triangle GIC is 4, find the area of triangl€ ,4rG D (D) 61 (A) 32 (B) 36 (c) 40 (E) 76 7. Which of ihe following five numbers has the greatest valu€? (A) (10 r)(10+r) (B) (11 ,r)(e+?.) (c) (12-r)(8+n) (D) (13 n)(7+n) (E) (14 ?l)(6+?r) Two real numbers z and t) satisfy the following €quations r€spectively: 2015?l'+2016u+1:0; ,'z+ zot6, +:ot: = o. II ar I L, find the value of I lAr 2016 19, I 2415 2015 (c) :oro! (l) :orsry (E) I L Find the minimum value of the real valued function z + 2016 - r,47 J. (A) 2015 (B) 2016 (C) 2017 (D) 201s (E) 201e The number 12345678S101112 . . . is formed by $.riting the $hole numbers 1,2,3, . . . until there are-201 digits in the number. Find the rcmainder when this number is dn ided bv 9. (A) 1 (B) 2 (c) 3 (D) 4 (E) 5 Short Questions If the sum and product of two positiwe real numbers ar.c both equal to 13, find the sum of the squares of these hvo numbers. Lel u b e a positive integer. .lf the highest coElmon factor of u and 168 is 12 and the hishesr common faclor of j. and 270 is 18, find the smallest possible value of z. 10. |. 12

13. In the diaeram, ,4-B, AC, C D ar'd DB arc tespectively the diameten of the circles q j c2, ca and c4. If the circumfer€nce of q is 2016, what is the sum of the circumferences of al] four circles? 14. Find the integer closest to 10./0J + 10!4J. 15. In the figure below, each distinct,letter represents a unique digit such that the arithmetic sum hoids. What is the five digit number represented by SIXTY? , F O R.. T Y TEN + TEN SIXTY. 16. In the diagram, ,4BCD is a rcctangle in which,4B :34 and BC :12. P al:d Q arc points on ,4.B and dD respectively such that ICPQ an.l IPQA are dght a,Irgtes. Find the suln of the two possible lensths of side ,4P. 17. Let a, b and c be positive irtegels such th:.t db + ac: L44 ab + bc : 209 ac + bc:221. Find the va.lue of a2 + b2 + c?.

18. 19. 20. P and Q are two regulax polygons with respectively ?, and m number of sides. The ratio of the inlerior angles of P ard Q is 4 : 3. If n > ra, hoe- many possible pairs of (rz, rn) are there? It is given ihat r and , aie positive integels such that f > g and /i r g '4o,tn. How many difi:erent possibl€ values can r can take? The diagram shoias a 12 by I rectangle which is cut by a pail of parallel line segmcxts into tbree parts with the equal areas. lf h denotes the distance betr,veen the two parallel lines, lind the value of 3ilr2. An examinatjon comp ses t$'o pape$ each with a tolal of 100 marks. In order to pass the examination, a candjdaLe must score at least 45 marks in each paper and al least 100 marks on th€ two papers combincd. Only integer marks will be given for each paper. Find the number of possible ways in which a candidat€ scores at least 45 marks in each paper and yet fails the examination. Find the sum of all the possible thrce digit numb€rs dr. srr,l, that the six digit number 741dbc is divisible bJ' 6, 7 and 10. ,4 and B are two right circular cylinders. 1'he curr.ect surlace area of,4 is 12.5% more than that of B whilc the bdse a,rea of A is 19% less than that of B. If the height of ,4 is a% more than that of B. find the value of u. ,!", 6"6 1L'" *1o" .1 f rj. 25. Find the number of pairs of positive integers " and g \,hich saiisl/ the equalion 2r, 21 24H 2b 3c 3r+\it=2016.

26. The diagram shows a seiies of inscdbed squares. The axea of the largesl outer square ABCD is 5I2. The fiIst inner square is ,41Brc1tl where ]{Ar : IAB, BB : lBC ' Cq : +CD and Dr1 : ]-o,4. The second inner square is AzBzCzDz where,4lA2: jl,ra|, A1A2: ]81C1 and so on. The third inner square which is the smallest in the diagram is formed in a similar 'ay Find the a-rea of the smallest square 28. Let r,!/ and z be positive integers that satisfy the equations r i y2 z: 100. n2+9r+20 I 270 27. Lel u,n2,r4,...,ns be nine distinct positive intege$ such that 7r1 < n2 < n3 < . < ne and n1 + n: + n3 +... + n9 : 180. Suppose that tbe \due of nr +n2+n3+n4+n5 is maximum. find the maximurn possiblc value of ne nr. 12 +y z:124 and Findthevalueofr+g+2. 29. Find the positive integer ,? such that 11 n2 Sn jb n.rln 2 30. 31. ,4EC is an isosceles triangle with lB : ,4C = 3. There are A distincl points on BC' denoiedbyPr,Pr,..., Pk.Letai:AP2+EP PiC wherei,:f,2,.. ,A. Fnld the mlua of k itur *rzi... *rr = 1080. If d and b are intege$ au,i rA - zu4 ;. ott" of thc roots of the equati ott * + ar + b:0' finil the value of a - b. 31. I rnd rlF -rsll".t n,pgFr r ru.i rb"l 2". (+s '4oro) > r.

33. The diagmm shows :,n isosceles t angle ABC with BA: BC and IABC : 100' ll D is a point on 8.4 produced such that BD = ,4C, flLd IBDC 34. Let c,g and z be positive real numbers such that c+y + z:1 Find the va.lue of 41 16s , 642 ltrt t l''ll Find the sum of a.]l positive iniegem n such that n'z + n + 2016 is aJ'd rg +lz + zx: 1 a pefect squore. 35.